Christoffel Formula

Let $\{p_n(x)\}$ be orthogonal Polynomials associated with the distribution $d\alpha(x)$ on the interval $[a,b]$. Also let

$$\rho\equiv c(x-x_1)(x-x_2)\cdots (x-x_l)$$

(for $c\not=0$) be a Polynomial of order $l$ which is Nonnegative in this interval. Then the orthogonal Polynomials $\{q(x)\}$ associated with the distribution $\rho(x)\,d\alpha(x)$ can be represented in terms of the Polynomials $p_n(x)$ as

$$\rho(x)q_n(x)=\left\vert\matrix{ p_n(x) & p_{n+1}(x) & \cdo... ..._n(x_l) & p_{n+1}(x_l) & \cdots & p_{n+l}(x_l)\cr}\right\vert.$$

In the case of a zero $x_k$ of multiplicity $m>1$, we replace the corresponding rows by the derivatives of order 0, 1, 2, ..., $m-1$ of the Polynomials $p_n(x_l)$, ..., $p_{n+l}(x_l)$ at $x=x_k$.

References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 29-30, 1975.